Civil and transportation engineers must often estimate the future traﬃc ﬂow on roads and bridges to plan for maintenance or possible future expansion. The following data gives the number of vehicles (in millions) crossing a bridge each year for 10 years. Fit a cubic polynomial to the data and use the ﬁt to estimate the ﬂow in year 2000.
Year 1990 1991 1992 1993 1994 Vehicle ﬂow (millions) 2.1 3.4 4.5 5.3 6.2
Year 1990 1991 1992 1993 1994 Vehicle ﬂow (millions) 6.6 6.8 7 7.4 7.8

Determine the value of the integral
I =
Z 1
0
x
p
1−x
2 dx
by composite trapezoidal rule with 3 and 5 ordinates. Improve the result by using
extrapolation technique.

The interpolating polynomial of degree \\(\\leq n\\)with the nodes\\(x_{0}, x_{1},\\cdots, x_{n}\\) can be written as

9.One of these describes the Lagrange’s interpolating polynomial P(x)
a.\\(p(x) = L_{1}(x)f_{0} + L_{2}(x)f_{1} + L_{3}(x)f_{2} + L_{4}(x)f_{3}\\)
b.\\(P(x) = L_{0}(x)f_{0} + L_{1}(x)f_{1} + L_{2}(x)f_{2} + L_{3}(x)f_{3}\\)
c.\\(P(x) = L_{0}(x)f_{3} + L_{1}(x)f_{2} + L_{2}(x)f_{1} + L_{3}(x)f_{0}\\)
d.\\(P(x) = L_{0}(x)f_{3} + L_{1}(x)f_{2} + L_{2}(x)f_{1} + L_{3}(x)f_{0}\\)
10.One of these is a method of solving system of linear equations:
a.Square Method
b.Inverse method
c.Lagrange Method
d.Transformation Method

7.Iterative methods of the solutions of systems of equations are ________________
a.finite
b.sequential
c.infinite
d.non-sequential
8.Stirling\'s formula for interpolation is given by
a.\\(P_n (x) = f (x_0) + \\frac{s}{2}[df_\\frac{1}{2} + d_\\frac{-1}{2}] + \\frac{s^2}{2}d^2f_0\\)
b.\\(P_n (x) = f (x_0) + \\frac{s}{2}[df_\\frac{1}{2} + d_\\frac{-1}{2}] + \\frac{s^2}{2!}d^2f_0\\)
c.\\(P_n (x) = f (x_0) + \\frac{s - 1}{2}[df_1+ d_\\frac{-1}{2}] + \\frac{s^2}{2!}d^2f_0\\)
d.\\(P_n (x) = f (x_0) + \\frac{s}{2}[df_\\frac{1}{2} + d_\\frac{-1}{2}] + \\frac{s^2}{2!}df_0\\)

5.The technique of determining an approximate value of f(x) for a non-tabular value of x which lies in the internal [a, b] is referred to as
a.interpolation
b.extrapolation
c.exterpolation
d.intrapolation
6.If all the elements above the main diagonal of a square matrix vanish, the matrix is known as
a.side triangular
b.half triangular
c.lower triangular
d.upper triangular

3.The zeroeth divided difference of the function f, with respect to \\( x_{i} \\) denoted by \\( f[x_{i}]\\) can be written as
a.\\(f[x] = f(xi)\\)
b.\\(f[x_{0}] = f(x_{i})\\)
c.\\(f[x] = f(x_{i})\\)
d.\\( f[x_{i}] = f(x_{i})\\)
4.When the number of rows is equal to the number of columns in a matrix, it is called _________ matrix
a.square
b.equal
c.similar
d.singular

1.The interpolating polynomial of degree \\(\\leq n\\)with the nodes\\(x_{0}, x_{1},\\cdots, x_{n}\\) can be written as
a.\\( f[x_{i}] = f(x_{i})\\)
b.\\( f[x_{i}] + f(x^i)=1\\)
c.\\( f[x_{i}] - f(x_{i}=0)\\)
d.\\( f[x_{i}] +f(x_{i}=0)\\)
2. A matrix in which all the non-diagonal elements vanish is called ___________ matrix.
a.diagonal
b.vanishing
c.non-diagonal
d.non-vanishing

An experiment to compare the lifespan of four different brands of spark plug was carried out. Five spark plugs of each brand were used, and the number of miles used until failure as an indicator of lifespan was Following is part of the statistical results for a one-way ANOVA.
ANOVA Table
Source of variation
Sum of squares
df
Mean square error
F-value
Treatments
176.5
B
E
G
Error
A
C
F
Total
236.0
D
Find the missing values for A, B, C and D in the ANOVA table.
Perform the ANOVA test to check for significant differences among the population means of the four different brands of spark plugs. Use a 0.05 significance level to conduct the test. The test should include the following steps:
a. State the null hypothesis and alternative hypothesis.
b. Specify the significance level and rejection region.
c. Determine the value of test statistic F. [
d. Determine test decision regarding the null hypothesis.
e. State your conclusion